Microscopic determination of the hole-motion-renormalized exchange $J_{\mathrm{eff}}=J-Ktx$, its chiral loop terms, and the fate of the flux/$s+id$ RVB pseudogap state in the doped cuprate Mott insulator

Anderson (2002) proposed that the scale of the pseudogap phase of the hole-doped cuprates — the crossover temperature $T^*$ at which the pseudogap opens — is set by the antiferromagnetic exchange of the spin sector of the doped Mott insulator, renormalized downward by fast hole motion: $J_{\mathrm{eff}}=J-Ktx$, with $x$ the hole concentration, $t$ the hopping, and $K$ an undetermined dimensionless constant of order unity (estimated $\approx 1$). The derivation — a Born-Oppenheimer-like trace over hole paths in the spirit of the Brinkman-Rice moment treatment of the one-hole (Nagaoka) problem — leaves three coupled pieces undetermined: the microscopic value of $K$ and the higher-order-in-$x$ corrections; the current-spin-current couplings generated by looped hole paths — the source paper's collective term for this loop-generated class (its Eq. (4)), whose one explicit member is the scalar-chirality (chiral three-spin) coupling — suspected to feed a staggered-flux (d-density-wave) phase; and the mechanism by which hole-motion fluctuations destabilize the flux/$s+id$ RVB fixed point at finite $x$ in favor of superconductivity, which the source paper deferred to a forthcoming paper that never appeared. This problem asks for the controlled effective spin Hamiltonian of the two-dimensional $t$-$J$ model at small finite $x$ with $t\gg J$ and the resulting fate of the flux/$s+id$ RVB state. The microscopic value of $K$, the loop-generated current-spin-current couplings, and the fate of the flux/$s+id$ state remain undetermined in the literature.

Background

The hole-doped cuprate superconductors (for example $\mathrm{Bi_2Sr_2CaCu_2O_{8+\delta}}$ or $\mathrm{YBa_2Cu_3O_{6+\delta}}$) are quasi-two-dimensional materials whose $\mathrm{CuO_2}$ planes, at one hole per Cu site, form a Mott insulator: an antiferromagnetic insulator whose insulating character arises from strong on-site Coulomb repulsion $U$ between electrons rather than from a filled band. The standard low-energy description is the one-band Hubbard model — the lattice model of correlated electrons with hopping $t$ and on-site repulsion $U$ — or, after eliminating double occupancy order by order in $t/U$, the $t$-$J$ model,

$$H = P\sum_{ij\sigma} t_{ij}\, c^\dagger_{i\sigma} c_{j\sigma} P + \sum_{ij} J_{ij}\,\mathbf{S}_i\cdot\mathbf{S}_j,$$

where $P$ projects onto states with no double occupancy, $t\approx 0.3$–$0.4\ \mathrm{eV}$ is the nearest-neighbor hopping amplitude, $J\approx 0.1\ \mathrm{eV}$ is the resulting antiferromagnetic superexchange coupling (the exchange generated by virtual hopping processes), and $x$ is the concentration of doped holes. Hopping dominates, $t\gg J$, and doping $x>0$ melts the Néel order (the conventional staggered antiferromagnetic pattern) and produces superconductivity below a dome-shaped $T_c(x)$.

The central unexplained feature of the underdoped phase diagram is the pseudogap: below a crossover line $T^*(x)$ the density of states at the Fermi level is partially depleted, antiferromagnetic correlations weaken, and angle-resolved photoemission shows truncated Fermi arcs instead of a full Fermi surface. Thermodynamic analysis (Tallon and Loram, 2001) finds $T^*(x)$ falling approximately linearly with doping and extrapolating to zero near $p^*\approx 0.19$, while high-field penetration-depth measurements (Shibauchi et al., 2001) independently find the pseudogap closing near $x\approx 0.2$, where the superconducting dome is highest. The microscopic origin of this scale is a leading open question of the field; candidate mechanisms include preformed Cooper pairs, hidden density-wave order, and fractionalization.

The source paper (Anderson, 2002) approaches $T^*$ from the resonating valence bond (RVB) theory of the doped Mott insulator (Anderson, 1987; Baskaran and Anderson, 1988). In RVB theory the spins of the parent antiferromagnet form a quantum liquid of singlet pairs (valence bonds) instead of static Néel order, and doping fractionalizes the electron into a neutral spin-$1/2$ spinon and a charged holon. In the fermionic slave-particle mean-field treatment — an auxiliary representation in which the physical electron operator is factorized into spinon and holon variables, with a projection (the Gutzwiller projection) restoring the no-double-occupancy constraint at the end — the exchange term becomes a pairing problem for spinons whose lowest translationally invariant solution is the flux phase or $s+id$ RVB state: a spinon pair condensate with a complex superposition of $s$- and $d$-wave channels, gauge-equivalent to spinons hopping in a background of $\pi$ flux per plaquette. This state carries Dirac-like gapless nodal spinons at $(\pm\pi/2,\pm\pi/2)$ and was shown (Hsu, 1990) to be unstable at $x=0$ to spinon pairing into magnons, i.e. to the antiferromagnet.

Two classical results about one-hole motion underpin the source paper's derivation. The Nagaoka theorem (Nagaoka, 1966): in the $U=\infty$ Hubbard limit a single hole in a fully ferromagnetic spin background moves coherently with bandwidth $\sim 4t$, and the ground state is ferromagnetic; in a generic spin background hole motion is obstructed because the hole leaves behind a trail of permuted spins. The Brinkman-Rice moment method (Brinkman and Rice, 1970) computes the one-hole spectrum from the local single-particle Green's function expanded in moments (traces of powers of the Hamiltonian on a fixed spin configuration) and finds that roughly three quarters of the one-hole bandwidth is independent of the spin configuration (hole paths that go out and back), the remainder coming from closed loops (single-plaquette circuits and longer) that act as a spin-configuration-dependent hole self-energy.

The source paper's construction — its title's umklapp (scattering that transfers crystal momentum to the lattice) marking the doping range where the spin sector's effective Hamiltonian governs observables — is a Born-Oppenheimer-like trace: since $t\gg J$, the fast holes are traced over with the slow spins frozen, as fast electrons generate an effective interaction for slow phonons. To first order in $x$ the loop self-energy is ferromagnetic (hole energies are negative and odd loops produce odd spin permutations), giving the renormalized exchange

$$J_{\mathrm{eff}} = J - Ktx,$$

with $K$ an unknown constant of order unity, estimated $\approx 1$ by comparison with the $4t$ free-hole bandwidth in a saturated background. For cuprate parameters the zero of $J_{\mathrm{eff}}$ falls between $x=0.2$ and $0.3$, tracking $T^*$. A single-plaquette loop also generates a chiral three-spin coupling proportional to $(\boldsymbol\sigma_1\times\boldsymbol\sigma_2)\cdot\boldsymbol\sigma_3$ — the scalar chirality of three spins — which cancels in the first-order average over the two circulation senses of the hole motion, but is suspected to be important for the staggered flux phase: a pattern of circulating bond currents alternating in sign between $x$- and $y$-directed bonds, breaking time-reversal and translation symmetry, closely related to the $d$-density-wave hidden order proposed as a rival account of the pseudogap (Chakravarty, Laughlin, Morr, and Nayak, 2001). With the chiral and higher-order terms dropped, $J_{\mathrm{eff}}\sum_{\langle ij angle}\mathbf{S}_i\cdot\mathbf{S}_j$ supports the flux/$s+id$ RVB state as an unstable fixed point — a stage through which the renormalized description passes before flowing elsewhere: at $x=0$ toward antiferromagnetic order, and at finite $x$, by assertion, toward superconductivity driven by the kinetic-energy terms at higher powers of $x$. The source paper defers that destabilization calculation to a forthcoming paper, which is the origin of the open problem stated below.

Problem Statement

Regime and object. Consider the two-dimensional $t$-$J$ model on the square lattice,

$$H = P\Big(\sum_{\langle ij\rangle\sigma} t_{ij}\, c^\dagger_{i\sigma} c_{j\sigma}\Big)P + \sum_{\langle ij\rangle} J_{ij}\,\mathbf{S}_i\cdot\mathbf{S}_j,$$

with $P$ the projector onto the subspace with no doubly occupied sites, $J\simeq t^2/U$, $t\gg J$, and hole concentration $x$ small but finite (the pseudogap regime $x\sim 0.05$–$0.25$ of the cuprates). Treating the hole motion as fast relative to the spin dynamics, define the effective spin Hamiltonian $H_{\mathrm{spin}}(x)$ by tracing over the hole degrees of freedom with the spin configuration held fixed (the Born-Oppenheimer-like procedure of the source paper). The problem is the controlled determination of $H_{\mathrm{spin}}(x)$ and of its consequences, in three coupled parts:

(1) The coefficient $K$ and higher orders in $x$. The source paper's first-order result is $J\to J_{\mathrm{eff}}=J_{\mathrm{super}}-Ktx$ with $K$ an unknown constant of order unity. Compute $K$ microscopically from the loop self-energy of a dilute gas of holes moving in a general (not saturated) spin background, determine the corrections at higher powers of $x$ (multi-hole terms, longer closed loops, hole-hole and hole-spin interactions), and state explicitly the regime of validity of the leading expression.

(2) The chiral and current-spin-current couplings. A single-plaquette hole loop generates, besides the ferromagnetic exchange correction, a scalar-chirality coupling proportional to $(\boldsymbol\sigma_1\times\boldsymbol\sigma_2)\cdot\boldsymbol\sigma_3$; the chirality cancels only after averaging over the two circulation senses of the hole motion at first order. Determine which symmetry-allowed couplings survive at finite $x$ at each order in the trace, and establish whether the chiral channel favors staggered-flux ($d$-density-wave) or related broken-symmetry order — the possibility the source paper suspected but could not settle.

(3) The fate of the flux/$s+id$ RVB state. With the chiral and higher-order terms dropped, $H_{\mathrm{spin}}=J_{\mathrm{eff}}\sum_{\langle ij\rangle}\mathbf{S}_i\cdot\mathbf{S}_j$ places the spin sector at the flux/$s+id$ RVB fixed point with nodal spinons — the candidate pseudogap state. Establish the mechanism by which the fluctuations induced by hole motion destabilize (or fail to destabilize) this state at finite $x$ and lower temperature — the calculation the source paper deferred to a forthcoming paper — and determine whether the kinetic-energy terms at higher powers of $x$ drive the superconducting instability, as the source paper asserted.

Determine $H_{\mathrm{spin}}(x)$ with quantitatively determined couplings ($K$ and the chiral and current-spin-current terms) within the approximation valid at small finite $x$ with $t\gg J$, and determine from those couplings the low-temperature phase of the doped Mott insulator: the pseudogap fixed point, whether the hole-motion fluctuations destabilize it, and — if they do — the channel and the instability that replaces it. The three parts are components of one objective — a controlled microscopic theory of the doped Mott insulator's spin sector within the source paper's program — since the couplings of (1) and (2) are precisely the inputs that decide (3).

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